Is it possible to beat the main story of GTA V with never having the character you are currently playing as kill someone?
No, it's not possible.
Sorry about that.
There are missions in the main story where there is no alternative to killing people. Listing them all would be futile, but an example as Michael in the By the Book mission.
Unfortunately not. Streamer DarkViper has attempted this using a vast amount of knowledge from speedrunning the game. His strategies include letting unused characters get kills, getting police to kill people and insane cheesy strategies that in one case took 33 hours to beat one mission.
He does not use the stun gun as, despite its name, it kills people. Even if it didn't, it's not available until after some of the mandatory kills (the prologue guard for example).
As of Meltdown, his killcount is 98 (59 melee, 28 gunshot, 1 samsung).